A.1 Annotated Bibliography of Print Resources

Glenn Elert

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A.1 Annotated Bibliography of Print Resources

There are thousands of printed resources on chaos, fractals, and dimension. These are the sources that inspired me to write this book.

Barnsley, Michael. Fractals Everywhere. San Diego, CA: Academic Press, 1988. If you want to really learn about fractals, this is the textbook I recommend. Easy to read.
Devaney, Robert L. Overview: Dynamics of Simple Maps. Chaos and Fractals: The Mathematics Behind the Computer Graphics. Robert L. Devaney & Linda Keen, eds. Providence, RI: American Mathematical Society, 1988.
Dewdney, A.K. Mathematical Recreations: Leaping into Lyapunov Space. Scientific American. September 1991: 178–180.
Feigenbaum, Mitchell J. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics. 19 (1978): 25–52. The paper that introduced the Feigenbaum constants, although they were not called this by the author.
Gleick, James. Chaos: Making a New Science. New York: Viking, 1987. More of a history lesson than a mathematics lesson. The companion DOS software package (which I have not reviewed) was available as shareware from San José State University for awhile.
Harrison, Jenny. An Introduction to Fractals. Chaos and Fractals: The Mathematics Behind the Computer Graphics. Robert L. Devaney & Linda Keen, eds. Providence, RI: American Mathematical Society, 1988, 107-126. A technical and readable introduction to fractals. The author mentions a study done by Tricot on "12 definitions of dimension".
- Tricot, Claude. Douze definitions de la densité logarithmique. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 293 (1981), 549–552.
- Tricot, Claude. Two [sic?] Definitions of Fractional Dimension. Mathematical Proceedings of the Cambridge Philosophical Society. 91 (1982): 57–74. That has to be a typo. Doesn't "douze" mean "twelve" not "two"?
Hocking, John G. & Young, Gail S. Topology. New York: Dover, 1961. Gail S. Young was my topology professor at Columbia University. The Dover reprint of this textbook is wonderfully inexpensive. I can easily recommend it for its price alone (something around US $9 in 1995).
Hofstadter, Douglas R. Metamagical Themas: Questing for the Essence of Mind and Pattern. New York: Basic Books, 1985. This was my first introduction to the world of chaos. The description is amazingly simple, but the conclusions are profound. By just goofing around with the parabola, one can generate an entirely new field of mathematics. How amazing is that?
Hurewicz, Witold & Henry Wallman. Dimension Theory. Princeton, NJ: Princeton University Press, 1941. References to this book appeared throughout my research. I have not read it. It was out of print for awhile and hard to find.
Hubbard, John Hamal (Dynamical Systems Laboratory). Mandelbrot Sets and Julia Sets. Videocassette. Ithaca, NY: Art Matrix: 1990. Now available in 10 parts on YouTube. Two hours of supercomputed fractal animation with a soundtrack that varies from new age to techno. No discussion or commentary, just animations. Includes Mandelbrot zooms, Julia promenades, cascade maps, and the Lorenz attractor. This videocassette is probably out of print. The distributor doesn't even sell fractal products anymore. (Fractals just weren't a good money maker I was told.) A somewhat obsolete product given that nearly all of these images can now be generated on a personal computer.
Jones, Jesse. Fractals for the Macintosh. Corte Madera, CA: The Waite Group, 1993. An inexpensive little book and CD combination that is now out of print. Fractals for the Macintosh disappeared along with the company that published it.
Keen, Linda Julia Sets. Chaos and Fractals: The Mathematics Behind the Computer Graphics. Robert L. Devaney & Linda Keen, eds. Providence, RI: American Mathematical Society, 1988, 57–74.
Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. A three volume encyclopedia of mathematical history.
Lauwerier, Hans. Fractals: Endlessly Repeated Geometrical Figures. translated by Sophia Gill-Hoffstädt. Princeton, NJ: Princeton University Press, 1991.
Lorenz, Edward N. Deterministic Nonperiodic Flow. Journal of Atmospheric Science. 20 (1963): 130-141. The paper that introduced us to the butterfly effect (although the term had not yet been coined).
Mandelbrot, Benoit B. The Fractal Geometry of Nature. revised edition. New York: W. H. Freeman and Company, 1977. "Clouds are not spheres, mountains are not cones, and lightening does not travel in a straight line." The book that introduced the Mandelbrot set (called the mu-set). Wanders around a bit, but very entertaining. Hundreds of examples included.
Markus, Mario. Chaos in Maps with Continuous and Discontinuous Maxima. Computers in Physics. September/October 1990: 481-493. The original article on the Lyapunov space diagrams featured in Chapter 4.4.
Nicolis, Grégoire & Prigogine, Ilya. Exploring Complexity: An Introduction. New York: Freeman, 1989.
Peitgen, H.O., H. Jürgens, D. Saupe, & C. Zahlten. Fractals: An Animated Discussion. Videocassette. New York: Freeman, 1990. A good non-technical overview of chaos and fractals featuring interviews with Mandelbrot and Lorenz. Their discussions are somewhat animated, but the narrator speaks like a hypnotist. (The chaotic background music doesn't help in this matter.) Not suitable for people with short attention spans. I liked it however.
Penrose, Roger. The Emperor's New Mind. New York: Oxford University Press, 1989.
Pickover, Clifford A. Chaos in Wonderland: Visual Adventures in a Fractal World. New York: St Martin's, 1995. Relevant excepts from this book can be found in the chapter The 15 Most Famous Transcendental Numbers.
Pickover, Clifford A. The World of Chaos. Computers in Physics. September/October 1990: 460–487.
Symon, Keith R. Mechanics. 3rd edition. Reading, MA: Addison Wesley, 1971. My undergraduate mechanics textbook and the resource I used to remind myself of all the mathematics behind the harmonic oscillator.